Group #14527930 — subgroups

Subgroups — #14527930

Order 2176.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763525	C₂	1	1	yes	maximal, characteristic
2	#11763188	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763100	C₂	1	2	yes	maximal, complemented
4	#11762536	C₂	1	2	yes	maximal, complemented
5	C₁₇×#333	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	#10587633	—	2	2	no	—
8	C₂×#63231	—	2	2	no	—
9	#10587581	—	2	2	no	—
10	#10587578	—	2	2	no	—
11	C₄×D₆₈	C₂²	1	2	yes	—
12	#10587555	C₂²	1	1	yes	characteristic
13	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
14	#10587492	—	2	2	no	—
15	#10587488	—	2	2	no	—
16	C₄×#3622	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₇	D₄	1	2	yes	—
18	C₂×D₆₈	—	2	2	no	—
19	C₂×D₆₈	D₄	1	2	yes	—
20	C₂×C₄×D₁₇	—	2	2	no	—
21	C₂×Dic₃₄	—	2	1	no	—
22	C₂×C₈×C₁₇	—	2	2	no	—
23	C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₄²×C₁₇	C₂³	1	1	yes	characteristic
25	#63231	—	4	4	no	—
26	#63227	—	2	4	no	—
27	#63227	—	2	2	no	—
28	C₄⋊Dic₁₇	—	2	2	no	—
29	C₄⋊Dic₁₇	D₄	1	2	yes	—
30	C₄×Dic₁₇	—	2	2	no	—
31	C₂×#3622	—	2	1	no	—
32	C₂²×D₁₇	—	2	2	no	—
33	Q₈×C₁₇	—	4	2	no	—
34	C₂×C₄×C₁₇	—	2	2	no	—
35	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
37	C₂×Dic₁₇	—	2	2	no	—
38	C₂×Dic₁₇	—	4	1	no	—
39	D₆₈	—	2	2	no	—
40	D₆₈	—	2	2	no	—
41	D₆₈	—	8	2	no	—
42	C₄×D₁₇	—	4	2	no	—
43	C₄×D₁₇	—	4	2	no	—
44	Dic₃₄	—	4	2	no	—
45	C₈×C₁₇	—	4	2	no	—
46	#3622	—	2	2	no	—
47	#1634	—	17	1	no	maximal, Sylow
48	C₂²×C₁₇	#120	1	1	yes	characteristic
49	C₂×D₁₇	—	4	2	no	—
50	C₂×D₁₇	—	4	2	no	—
51	C₂×D₁₇	—	4	2	no	—
52	C₄×C₁₇	—	2	1	no	—
53	C₄×C₁₇	—	4	2	no	—
54	C₄×C₁₇	#135	1	2	yes	—
55	C₄×C₁₇	#137	1	2	yes	—
56	Dic₁₇	—	4	2	no	—
57	Dic₁₇	—	8	1	no	—
58	#555	—	17	1	no	—
59	#494	—	17	1	no	—
60	#476	—	17	2	no	—
61	#333	D₁₇	1	1	yes	characteristic, complemented
62	D₄⋊C₈	—	17	2	no	—
63	C₂×C₁₇	#452	1	2	yes	—
64	C₂×C₁₇	#453	1	1	yes	characteristic
65	D₁₇	—	8	2	no	—
66	C₂×Q₁₆	—	34	2	no	—
67	C₄⋊Q₈	—	17	1	no	—
68	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
69	#124	—	34	2	no	—
70	#121	—	34	2	no	—
71	C₄×D₄	—	17	2	no	—
72	#117	—	34	2	no	—
73	#106	—	34	2	no	—
74	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
75	#102	—	34	2	no	—
76	C₄×C₈	—	17	1	no	—
77	C₁₇	#1634	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₈	—	34	1	no	—
79	C₂×Q₈	D₆₈	1	2	yes	—
80	C₂×D₄	—	17	2	no	—
81	C₂×D₄	—	34	2	no	—
82	C₂²×C₄	—	34	2	no	—
83	Q₁₆	—	68	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	34	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	17	2	no	—
88	C₄⋊C₄	—	34	2	no	—
89	C₂²⋊C₄	—	34	4	no	—
90	C₂²⋊C₄	—	34	2	no	—
91	C₄²	—	34	2	no	—
92	C₄²	C₂²×D₁₇	1	1	yes	characteristic
93	C₂³	—	34	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	68	2	no	—
96	D₄	—	34	2	no	—
97	D₄	—	34	2	no	—
98	D₄	—	136	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	34	2	no	—
101	C₂×C₄	—	68	2	no	—
102	C₂×C₄	—	68	2	no	—
103	C₂×C₄	—	68	1	no	—
104	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
105	C₂×C₄	D₄×D₁₇	1	2	yes	—
106	C₈	—	4	2	no	—
107	C₈	—	34	2	no	—
108	C₂²	—	68	2	no	—
109	C₂²	—	68	2	no	—
110	C₂²	—	68	2	no	—
111	C₂²	#10587568	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	68	2	no	—
115	C₄	—	136	1	no	—
116	C₄	#10587602	1	2	yes	—
117	C₄	#10587604	1	2	yes	—
118	C₂	—	136	2	no	—
119	C₂	#11762974	1	2	yes	—
120	C₂	#11762979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763525

C₂

yes

maximal, characteristic

#11763188

C₂

yes

maximal, characteristic, complemented

#11763100

C₂

yes

maximal, complemented

#11762536

C₂

yes

maximal, complemented

C₁₇×#333

C₂

yes

maximal, characteristic, complemented

C₁₇×C₄⋊Q₈

C₂²

yes

—

—

—

—

C₄×D₆₈

C₂²

yes

—

#10587555

C₂²

yes

characteristic

C₁₇×C₄⋊C₈

C₂²

yes

—

#10587492

—

#10587488

—

C₄×#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

D₄

yes

—

C₂×D₆₈

—

C₂×D₆₈

D₄

yes

—

C₂×C₄×D₁₇

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂³

yes

—

—

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

D₄

yes

—

C₄×Dic₁₇

—

C₂×#3622

—

C₂²×D₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

D₆₈

—

D₆₈

—

D₆₈

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

C₈×C₁₇

—

—

—

C₂²×C₁₇

yes

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#135

yes

—

C₄×C₁₇

#137

yes

—

Dic₁₇

—

Dic₁₇

—

—

—

—

D₁₇

yes

characteristic, complemented

D₄⋊C₈

—

C₂×C₁₇

#452

yes

—

C₂×C₁₇

#453

yes

characteristic

D₁₇

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

#124

—

#121

—

C₄×D₄

—

#117

—

#106

—

C₄⋊C₈

C₂×D₁₇

yes

—

#102

—

C₄×C₈

—

C₁₇

#1634

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

D₆₈

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

136

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₆₈

yes

characteristic

105

C₂×C₄

D₄×D₁₇

yes

—

106

C₈

—

107

C₈

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#10587568

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

136

—

116

C₄

#10587602

yes

—

117

C₄

#10587604

yes

—

118

C₂

—

136

—

119

C₂

#11762974

yes

—

120

C₂

#11762979

yes

characteristic

	subgroup	G/H
center	C₂²	#10587568
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#333	C₂
socle	C₂²×C₁₇	#120
prime radical	#11763525	C₂
hypercenter	#333	D₁₇
hypocenter	C₁₇	#1634
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#10587568

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#333

C₂

C₂²×C₁₇

C₂

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #11763188	C₂	semidirect
3. #11763100	C₂	semidirect
4. #11762536	C₂	semidirect
5. C₁₇×#333	C₂	semidirect
61. #333	D₁₇	semidirect
77. C₁₇	#1634	semidirect

H ≅ G/N

type

2. #11763188

C₂

semidirect

3. #11763100

C₂

semidirect

4. #11762536

C₂

semidirect

5. C₁₇×#333

C₂

semidirect

61. #333

D₁₇

semidirect

77. C₁₇

#1634

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11763525	2
#11763188	2
#11763100	2
#11762536	2
C₁₇×#333	2
#1634	17

subgroup

index

C₁₇×#333

Subgroups — #14527930

Conjugacy classes of subgroups

Complements

Maximal subgroups